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252

Aug 30, 2008
08/08

by
Ethan Smith

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Book digitized by Google from the library of the University of Michigan and uploaded to the Internet Archive by user tpb.

Source: http://books.google.com/books?id=H-I2AAAAMAAJ&oe=UTF-8

165
165

Aug 15, 2009
08/09

by
Ethan Smith

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Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb.

Source: http://books.google.com/books?id=oKIQAAAAYAAJ&oe=UTF-8

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24

Sep 22, 2013
09/13

by
Ethan Smith

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Let $K$ be a fixed number field, and assume that $K$ is Galois over $\qq$. Previously, the author showed that when estimating the number of prime ideals with norm congruent to $a$ modulo $q$ via the Chebotar\"ev Density Theorem, the mean square error in the approximation is small when averaging over all $q\le Q$ and all appropriate $a$. In this article, we replace the upper bound by an asymptotic formula. The result is related to the classical Barban-Davenport-Halberstam Theorem in the...

Source: http://arxiv.org/abs/1210.3863v1

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36

Sep 22, 2013
09/13

by
Ethan Smith

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For a fixed number field $K$, we consider the mean square error in estimating the number of primes with norm congruent to $a$ modulo $q$ by the Chebotar\"ev Density Theorem when averaging over all $q\le Q$ and all appropriate $a$. Using a large sieve inequality, we obtain an upper bound similar to the Barban-Davenport-Halberstam Theorem.

Source: http://arxiv.org/abs/1210.3862v1

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22

Sep 18, 2013
09/13

by
Ethan Smith

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Let $L/K$ be a Galois extension of number fields. The problem of counting the number of prime ideals $\mathfrak p$ of $K$ with fixed Frobenius class in $\mathrm{Gal}(L/K)$ and norm satisfying a congruence condition is considered. We show that the square of the error term arising from the Chebotar\"ev Density Theorem for this problem is small "on average." The result may be viewed as a variation on the classical Barban-Davenport-Halberstam Theorem.

Source: http://arxiv.org/abs/1201.6007v1

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221

Aug 24, 2008
08/08

by
Ethan Smith

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Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb.

Source: http://books.google.com/books?id=kTEAAAAAYAAJ&oe=UTF-8

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680

Feb 24, 2008
02/08

by
Ethan Smith

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Book digitized by Google from the library of the University of Michigan and uploaded to the Internet Archive by user tpb.

Source: http://books.google.com/books?id=thFZYLuCY_4C&oe=UTF-8

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41

May 30, 2012
05/12

by
Ethan Smith

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Student reading of popular children stories

Topic: If You Take a Mouse to School

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14

Sep 23, 2013
09/13

by
Kevin James; Ethan Smith

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Let $K$ be a number field and $r$ an integer. Given an elliptic curve $E$, defined over $K$, we consider the problem of counting the number of degree two prime ideals of $K$ with trace of Frobenius equal to $r$. Under certain restrictions on $K$, we show that "on average" the number of such prime ideals with norm less than or equal to $x$ satisfies an asymptotic identity that is in accordance with standard heuristics. This work is related to the classical Lang-Trotter conjecture and...

Source: http://arxiv.org/abs/1109.4007v2

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18

Sep 21, 2013
09/13

by
Chantal David; Ethan Smith

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Given an elliptic curve $E$ and a positive integer $N$, we consider the problem of counting the number of primes $p$ for which the reduction of $E$ modulo $p$ possesses exactly $N$ points over $\mathbb F_p$. On average (over a family of elliptic curves), we show bounds that are significantly better than what is trivially obtained by the Hasse bound. Under some additional hypotheses, including a conjecture concerning the short interval distribution of primes in arithmetic progressions, we obtain...

Source: http://arxiv.org/abs/1108.3539v3

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22

Sep 22, 2013
09/13

by
Kevin James; Ethan Smith

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Let $K$ be a fixed number field, assumed to be Galois over $\mathbb Q$. Let $r$ and $f$ be fixed integers with $f$ positive. Given an elliptic curve $E$, defined over $K$, we consider the problem of counting the number of degree $f$ prime ideals of $K$ with trace of Frobenius equal to $r$. Except in the case $f=2$, we show that "on average," the number of such prime ideals with norm less than or equal to $x$ satisfies an asymptotic identity that is in accordance with standard...

Source: http://arxiv.org/abs/1210.4603v1

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21

Sep 22, 2013
09/13

by
Vorrapan Chandee; Chantal David; Dimitris Koukoulopoulos; Ethan Smith

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It is well-known that if $E$ is an elliptic curve over the finite field $\F_p$, then $E(\F_p)\simeq\Z/m\Z\times\Z/mk\Z$ for some positive integers $m, k$. Let $S(M,K)$ denote the set of pairs $(m,k)$ with $m\le M$ and $k\le K$ such that there exists an elliptic curve over some prime finite field whose group of points is isomorphic to $\Z/m\Z\times\Z/mk\Z$. Banks, Pappalardi and Shparlinski recently conjectured that if $K\le (\log M)^{2-\epsilon}$, then a density zero proportion of the groups in...

Source: http://arxiv.org/abs/1210.3880v1

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Sep 22, 2013
09/13

by
Jessica F. Burkhart; Neil J. Calkin; Shuhong Gao; Justine C. Hyde-Volpe; Kevin James; Hiren Maharaj; Shelly Manber; Jared Ruiz; Ethan Smith

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In this paper, we recursively construct explicit elements of provably high order in finite fields. We do this using the recursive formulas developed by Elkies to describe explicit modular towers. In particular, we give two explicit constructions based on two examples of his formulas and demonstrate that the resulting elements have high order. Between the two constructions, we are able to generate high order elements in every characteristic. Despite the use of the modular recursions of Elkies,...

Source: http://arxiv.org/abs/1210.3504v1